Projection theorems for linear-fractional families of projections

Abstract

AbstractMarstrand’s theorem states that applying a generic rotation to a planar setAbefore projecting it orthogonally to thex-axis almost surely gives an image with the maximal possible dimension$\min(1, \dim A)$. We first prove, using the transversality theory of Peres–Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in$PSL(2,\mathbb{C})$or a generic real linear-fractional transformation in$PGL(3,\mathbb{R})$. We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of$PSL(2,\mathbb{C})$or$PGL(3,\mathbb{R})$. Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries.